Structural Boundary Design via Level Set and Immersed Interface Methods

A distributed Lagrange multiplier/fictitious domain method for particulate flows

This paper presents some efficient algorithms based on the Legendre–Galerkin approximations for the direct solution of the second- and fourth-order elliptic equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with sparse matrices for the discrete variational formulations. The complexities of the algorithms are a small multiple of $N^{d + 1} $ operations for a d-dimensional domain with $(N - 1)^d $ unknowns, while the convergence rates of the algorithms are exponential for problems with smooth solutions. In addition, the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix-matrix multiplications.

/pdf/efficient-spectral-galerkin-method-i-direct-solvers-of-gxmm0igvgl.pdf

Efficient spectral-Galerkin method I: direct solvers of second-and fourth-order equations using Legendre polynomials

A fictitious domain method for Dirichlet problem and applications

A second-order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the second-order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper [SIAM J. Numer. Anal., 31(1994), pp. 1019--1044]. The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasi-Newton method is developed that allows reasonable time steps to be used.

/pdf/immersed-interface-methods-for-stokes-flow-with-elastic-8qc9aoc7ob.pdf

Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension

There are several very fast direct methods which can be used to solve the discrete Poisson equation on rectangular domains. We show that these methods can also be used to treat problems on irregular regions.

/pdf/the-direct-solution-of-the-discrete-poisson-equation-on-5e6pqbsug0.pdf

The direct solution of the discrete Poisson equation on irregular regions

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The direct solution of the discrete Poisson equation on irregular regions